MONTREAL, CANADA – A significant breakthrough in spectral geometry has been achieved by a professor and his team at Université de Montréal. The team, led by Iosif Polterovich, has successfully proven a special case of Pólya’s conjecture related to the eigenvalues of a disk, solving a complex mathematical problem that has puzzled researchers for decades.
Using spectral geometry, a branch of mathematics, Polterovich and his collaborators delved into the question of whether it is possible to deduce the shape of a drum from the sounds it creates. This question led them to investigate the frequencies of a round drum, or the eigenvalues of a disk, as a means of understanding wave propagation phenomena.
In a collaboration with international mathematicians Nikolay Filonov, Michael Levitin, and David Sher, Polterovich was able to confirm Pólya’s conjecture for the disk, a feat that had previously eluded researchers. The conjecture, which was formulated by George Pólya in 1954, had only been proven for domains that tile a plane, such as triangles and rectangles, until last year.
Their findings, published in the mathematical journal Inventiones Mathematicae in July 2023, have shed light on the universal value and artistic beauty of mathematical research. While the result is primarily of theoretical significance, the team’s proof method has potential applications in computational mathematics and numerical computation, sparking further exploration in the field.
Polterovich emphasized the parallels between mathematics, sports, and the arts, noting that proving a conjecture is akin to a sport, while finding an elegant solution is an art. He highlighted the potential for beautiful mathematical discoveries to have practical applications, emphasizing the importance of finding the right context in which to apply them.
The research team’s work not only advances our understanding of spectral geometry but also underscores the interdisciplinary nature of mathematical research. Their success in proving Pólya’s conjecture for the disk marks a significant milestone in the field, showcasing the power of collaboration and innovative thinking in pushing the boundaries of mathematical knowledge.